Some non-commuting solutions of the Yang-Baxter-like matrix equation

Abstract

Let A be a square matrix satisfying {A}^{4}=A . We solve the Yang-Baxter-like matrix equation AXA=XAX to find some solutions, based on analysis of the characteristic polynomial of A and its eigenvalues. We divide the problem into small cases so that we can find the solution easily. Finally, in order to illustrate the results, two numerical examples are presented.

Explicit solutions of the Yang-Baxter-like matrix equation for a singular diagonalizable matrix with three distinct eigenvalues

“…Finding all non-commuting solutions of Yang-Baxter-like matrix equation ( 1) is still a challenging task when A is arbitrary. Up to now, there are only isolated results toward this goal for special classes of the given matrix A, e.g., [17][18][19][20][21][22][23][24][25][26][27][28]. All solutions have been constructed for rank-1 matrices A in [23], rank-2 matrices A in [24,25], non-diagonalizable elementary matrices A in [26], idempotent matrices A (A 2 = A) in [19], A 2 = I in [18,20], A 3 = A in [21], A 4 = A in [27], and diagonalizable matrices A with two different eigenvalues in [22].…”